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Manifolds with Aspherical Singular Riemannian Foliations

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Manifolds with Aspherical Singular Riemannian Foliations Manifolds with aspherical singular Riemannian foliations Zur Erlangung des akademischen Grades eines DOKTORSDERNATURWISSENSCHAFTEN von der Fakult¨at fur¨ Mathematik des Karlsruher Institut fur¨ Technologie (KIT) genehmigte DISSERTATION von Diego Corro Tapia Datum der mundlichen¨ Prufung:¨ 4. Juli 2018 Referent: Prof. Dr. Wilderich Tuschmann Korreferent: PD. Dr. Fernando Galaz-Garc´ıa KIT – Universit¨at des Landes Baden-Wurttemberg¨ und nationales Forschungszentrum der Helmholtz-Gesellschaft http://www.kit.edu/ Formated using: The ksp-thesis class. All figures created using Inkscape and TikZ. This work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license. Fur¨ meine Eltern A mis padres ABSTRACT In the present work we study A-foliations, i.e. singular Riemannian foliations with regular leaf aspherical. The main result is that, for a simply-connected closed (n + 2)- manifold M, an A-foliation with regular leaves of codimension 2 in M is homoge- neous. In other words it is given by a smooth effective action of the torus Tn on M by isometries. We will give some conditions to compare two simply-connected, closed manifolds with A-foliations, up to foliated homeomorphism, via their leaf spaces. i Film is one of the three universal languages, the other two: mathematics and music. — Frank Cappra Non est regia ad Geometriam via. — Euclid. ACKNOWLEDGMENTS During the research project which culminated in this thesis, I received support from several institutions and people. First of all I would like to thank my advisors Prof. Wilderich Tuschmann, and PD. Dr. Fernando Galaz-Garc´ıafor encouraging me to undertake this project. I am ever grateful to them for the discussions we had, as well as for sharing their knowledge and insight into the fascinating world of geometry. In particular I want to thank Prof. Tuschmann for providing a framework in which I was able to visit several mathematical institutions around the world. This had as a direct consequence, that I was able obtain a very panoramic vision of the field of geometry, as well as the opportunity to interact to several people who helped to enrich this work. To Fernando I am thankful for making a very open and creative work ambient at the institute, which facilitates the discussion and development of ideas. I also want to thank my other colleagues in the Differential Geometry Group at the KIT, Jan-Bernarhd Kordaß, Karla Garc´ıa,and Martin G¨unter, for the dis- cussions we had on common interests. In particular I want to thank Jan-Bernhard Kordaß for reading the drafts of this work, and sharing his comments, as well as for being an outstanding colleague. I also want to thank Catherina Campagnolo for sharing her knowledge and insight. There are special mentions to Prof. Luis Guijarro and Prof. Thomas Farrell, for pointing to directions that helped me point out the diffeomorphism types of the leaves. I want to thank also Prof. Alexan- der Lytchak for discussions which led to finding obstructions for the existence of a cross-section. During my research program I had the opportunity of visiting the University of Notre Dame. I am grateful to Professor Marco Radeschi, for hosting me and helping me to weed through the technicalities, at a point of the project when time was pressing. I also want to thank Professor Karsten Grove for discussion which led to a better presentation of the subject in this present work. I want to thank Adam Moreno for all the interesting discussions we had about foliations, geometry and life. iii iv acknowledgments In particular my I was able to incorporate a different view point of foliations from these discussions. Also thanks to the Institute Henri Poincar´efor providing a very nice library, where I had the pleasure to work for a few weeks. Last I want to thank all the people that lent support in some way during this doctoral research. First of all to Professor Oscar Palmas who suggested me to contact the group of Professor Tuschmann. I am also grateful for the conversations we have every time I go back to Mexico. Thanks to Dr. Jes´usNu˜nes-Zimbr´on,Dr. David Gonaz´alez-Alvaro,´ Dr. Masoumeh Zarei, Jaime Santos-Rodr´ıguez,Agustin Romano-Vel´azquezfor sharing their passion for geometry. To Jos´eLuis Cisneros for his support and the amazing basis of knowledge with which I started this project. To my friends who supported and encouraged me to keep going on during these last 3 years: Adri´an,Iker, David, Luigi, Citlali, Lau, Corinto, Irene, Sean and Gina, and Valerio. I am grateful to my brother Xavier for keeping me humble. Dedications go to Ana Lucia and Gaby for always believing in me. To my dad Leonardo and my mom Ailali for all their love and support. Special thanks to Rebeca for all her love, support, delicious food, and her patience, specially this last six months. The present doctoral work was developed and written under the support of CONACyT–DAAD Scholarship No. 409912. Karlsruhe, 4. Juni 2018 CONTENTS Abstracti Acknowledgments iii 1 Introduction1 I Background 11 2 Group Actions 13 2.1 Compact Lie Group Actions . 13 2.2 Orbit Types . 18 2.3 Torus Actions . 21 2.4 Isometric Actions . 28 3 Riemannian Foliations 33 3.1 Singular Riemannian foliations. 33 3.2 Infinitesimal foliation. 36 3.3 Holonomy and types of leaves . 39 3.4 Homogeneous foliations . 45 II Aspherical Foliations 49 4 Cross-sections and A-foliations 51 4.1 Cross-section for the leaf space . 51 4.2 A-foliations . 63 4.3 Molino Bundle . 66 4.4 Weights of an A-foliation . 71 5 A-foliations of codimension 2 79 5.1 Leaf space of A-foliations of codimension 2 . 79 5.2 Weights of A-foliation of codimension 2 . 82 5.3 Top. classification of A-foliations of codim. 2 . 83 6 Smooth Structure of Leaves of an A-Foliation 89 6.1 Fibrations between leaves . 90 6.2 Four dimensional torus. 91 6.3 Higher dimensional torus. 92 Appendix 99 A Linearized Flows 101 1 Linearized vector fields . 101 ReferencesI v Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. — Felix Klein 1 INTRODUCTION When studying a Riemannian manifold M, an approach to understand its geometry or its topology is to simplify the problem by “reducing” M to a lower dimensional space B. This can be achieved by considering a partition of the original manifold M into submanifolds which are, roughly speaking, compatible with the Riemannian structure of M. We then study the geometry or topology of B, with the aim of recovering information on M. As an example of this “reduction”, we can consider Riemannian submersions from M onto lower dimensional manifolds. We then study the properties of M which remain invariant along the fibers of the submersion. A concrete example of this is present in [GG87] and [Wil01]. The authors prove that a closed, simply- connected, Riemannian manifold M with sectional curvature greater or equal to 1, and diameter equal to π/2 is either homeomorphic to a sphere, or isometric to a compact symmetric space of rank one (a so called CROSS). As a key step in the proof, they show that any Riemannian submersion π : Sn → B onto some Riemannian manifold B is a Hopf fibration. This “reduction” approach is also present when we consider Riemannian mani- folds with an effective isometric action by a compact Lie group. In particular, this approach has been applied to the long-standing open problem in Riemannian geome- try, of classifying and constructing Riemannian manifolds of positive or nonnegative (sectional) curvature. Namely Grove has proposed in the symmetry program to first consider such manifolds with a high degree of symmetry, i.e. with an isometric action of a compact Lie group (see [Gro02]). 1 2 introduction The philosophy behind this approach is that by understanding first positively or nonnegatively curved manifolds with symmetry one may gain insight into the general case, either by constructing new examples or by finding possible obstructions. This has proved a successful approach, since many results have come to light by following loosely the symmetry program (see for example [Bre72], [Gro02], [Gro17], [Kob95], [Sea14],[Wil06]). This point of view has even been applied to other lower curvature bounds, such as positive Ricci (see for example [CGG16]), as it provides many tools and much flexibility. Since, in particular, any compact connected Lie group contains a maximal torus as a Lie subgroup, the study of torus actions is of importance in the study of group actions. The classification up to equivariant diffeomorphism of smooth, closed, simply-connected, manifolds with torus actions is a well studied problem when either the dimension of the manifolds or the cohomogeneity of the action is low (see for example [OR70],[KMP74],[Fin77],[Oh83a],[Oh82]). Both of these phenomena, Riemannian submersions and compact Lie group ac- tions, are encompassed in the more general concept of singular Riemannian foli- ations. In Riemannian geometry, singular Riemannian foliations have recently at- tracted the attention of many authors (see, for example, the survey [ABT13]) and led to many interesting results. Alexandrino has obtained information on the geometry of a manifold admit- ting certain types of singular Riemannian foliations, called polar foliations (see [Ale10, ABT13]). Singular Riemannian foliations have also led to results in dif- ferential topology, such as those surveyed in [QG16]. For example, one can obtain a lower bound on the number of distinct smooth structures a manifold with a singular Riemannian foliation can have. Also, as in the case of smooth effective torus actions, Radeschi and Ge obtained in [GR15] an explicit classification up to diffeomorphism of closed simply-connected 4-manifolds admitting a singular Riemannian foliation.
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